Canonical transformations map the n-dimensional anisotropic oscillator to the isotropic oscillator, establishing maximal superintegrability for commensurate frequencies with explicit first integrals in 2D.
In the following discussion, we suppress the index j and focus on two general functions X = X (X, P ) and P = P(X, P ), where we assume that P ̸ = 0
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Dynamical symmetries of the anisotropic oscillator
Canonical transformations map the n-dimensional anisotropic oscillator to the isotropic oscillator, establishing maximal superintegrability for commensurate frequencies with explicit first integrals in 2D.